### Jones inclusions and construction of S-matrix and U matrix

TGD leads naturally to zero energy ontology which reduces to the positive energy ontology of the standard model only as a limiting case. In this framework one must distinguish between the U-matrix characterizing the unitary process associated with the quantum jump (and followed by state function reduction and state preparation) and the S-matrix defining time-like entanglement between positive and negative energy parts of the zero energy state and coding the rates for particle reactions which in TGD framework correspond to quantum measurements reducing time-like entanglement.

**1. S-matrix**

In zero energy ontology S-matrix characterizes time like entanglement of zero energy states (this is possible only for HFFs for which Tr(SS^{+})=Tr(Id)=1 holds true). S-matrix would code for transition rates measured in particle physics experiments with particle reactions interpreted as quantum measurements reducing time like entanglement. In TGD inspired quantum measurement theory measurement resolution is characterized by Jones inclusion (the group G defines the measured quantum numbers), N subset M takes the role of complex numbers, and state function reduction leads to N ray in the space M/N regarded as N module and thus from a factor to a sub-factor.

The finite number theoretic braid having Galois group G as its symmetries is the space-time correlate for both the finite measurement resolution and the effective reduction of HFF to that associated with a finite-dimensional quantum Clifford algebra M/N. SU(2) inclusions would allow angular momentum and color quantum numbers in bosonic degrees of freedom and spin and electro-weak quantum numbers in spinorial degrees of freedom. McKay correspondence would allow to assign to G also compact ADE type Lie group so that also Lie group type quantum numbers could be included in the repertoire.

Galois group G would characterize sub-spaces of the configuration space ("world of classical worlds") number theoretically in a manner analogous to the rough characterization of physical states by using topological quantum numbers. Each braid associated with a given partonic 2-surface would correspond to a particular G that the state would be characterized by a collection of groups G. G would act as symmetries of zero energy states and thus of S-matrix. S-matrix would reduce to a direct integral of S-matrices associated with various collections of Galois groups characterizing the number theoretical properties of partonic 2-surfaces. It is not difficult to criticize this picture.

- Why time like entanglement should be always characterized by a unitary S-matrix? Why not some more general matrix? If one allows more general time like entanglement, the description of particle reaction rates in terms of a unitary S-matrix must be replaced with something more general and would require a profound revision of the vision about the relationship between experiment and theory. Also the consistency of the zero energy ontology with positive energy ontology in time scales shorter than the time scale determined by the geometric time interval between positive and negative energy parts of the zero energy state would be lost. Hence the easy way to proceed is to postulate that the universe is self-referential in the sense that quantum states represent the laws of physics by coding S-matrix as entanglement coefficients.
- Second objection is that there might a huge number of unitary S-matrices so that it would not be possible to speak about quantum laws of physics anymore. This need not be the case since super-conformal symmetries and number theoretic universality pose extremely powerful constraints on S-matrix. A highly attractive additional assumption is that S-matrix is universal in the sense that it is invariant under the inclusion sequences defined by Galois groups G associated with partonic 2-surfaces. Various constraints on S-matrix might actually imply the inclusion invariance.
- One can of course ask why S-matrix should be invariant under inclusion. One might argue that zero energy states for which time-like entanglement is characterized by S-matrix invariant in the inclusion correspond to asymptotic self-organization patterns for which U-process and state function reduction do not affect the S-matrix in the relabelled basis. The analogy with a fractal asymptotic self-organization pattern is obvious.

**2. U-matrix**

In a well-defined sense U process seems to be the reversal of state function reduction. Hence the natural guess is that U-matrix means a quantum transition in which a factor becomes a sub-factor whereas state function reduction would lead from a factor to a sub-factor.

Various arguments suggest that U matrix could be almost trivial and has as a basic building block the so called factorizing S-matrices for integrable quantum field theories in 2-dimensional Minkowski space. For these S-matrices particle scattering would mean only a permutation of momenta in momentum space. If S-matrix is invariant under inclusion then U matrix should be in a well-defined sense almost trivial apart from a dispersion in zero modes leading to a superpositions of states characterized by different collections of Galois groups.

**3. Relation to TGD inspired theory of consciousness**

U-matrix could be almost trivial with respect to the transitions which are diagonal with respect to the number field. What would however make U highly interesting is that it would predict the rates for the transitions representing a transformation of intention to action identified as a p-adic-to-real transition. In this context almost triviality would translate to a precise correlation between intention and action.

The general vision about the dynamics of quantum jumps suggests that the extension of a sub-factor to a factor is followed by a reduction to a sub-factor which is not necessarily the same. Breathing would be an excellent metaphor for the process. Breathing is also a metaphor for consciousness and life. Perhaps the essence of living systems distinguishing them from sub-systems with a fixed state space could be cyclic breathing like process N→ M supset N → N_{1} subset M→ .. extending and reducing the state space of the sub-system by entanglement followed by de-entanglement.

One could even ask whether the unique role of breathing exercise in meditation practices relates directly to this basic dynamics of living systems and whether the effect of these practices is to increase the value of M:N and thus the order of Galois group G describing the algebraic complexity of "partonic" 2-surfaces involved (they can have arbitrarily large sizes). The basic hypothesis of TGD inspired theory of cognition indeed is that cognitive evolution corresponds to the growth of the dimension of the algebraic extension of p-adic numbers involved.

If one is willing to consider generalizations of the existing picture about quantum jump, one can imagine that unitary process can occur arbitrary number of times before it is followed by state function reduction. Unitary process and state function reduction could compete in this kind of situation.

**4. Fractality of S-matrix and translational invariance in the lattice defined by sub-factors**

Fractality realized as the invariance of the S-matrix in Jones inclusion means that the S-matrices of N and M relate by the projection P: M→N as S_{N}=PS_{M}P. S_{N} should be equivalent with S_{M} with a trivial re-labelling of strands of infinite braid.

Inclusion invariance would mean translational invariance of the S-matrix with respect to the index n labelling strands of braid defined by the projectors e_{i}. Translations would act only as a semigroup and S-matrix elements would depend on the difference m-n only. Transitions can occur only for m-n≥ 0, that is to the direction of increasing label of strand. The group G leaving N element-wise invariant would define the analog of a unit cell in lattice like condensed matter systems so that translational invariance would be obtained only for translations m→ m+ nk, where one has n≥ 0 and k is the number of M(2,C) factors defining the unit cell. As a matter fact, this picture might apply also to ordinary condensed matter systems.

For more details see the chapters Construction of Quantum Theory: S-matrix and Was von Neumann Right After All? of "Towards S-matrix".

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